Recent work on emergence in physics has focused on the presence of singular limit relations between basal and upper-level theories as a criterion for emergence. However, over-emphasis on the role of singular limit relations has somewhat obscured what it means to say that a property or behaviour is emergent. This paper argues that singular limits are not central to emergence and develops an alternative account of emergence in terms of the failure of basal explainability. As a consequence, emergence and reduction, long held to be two sides of the same coin in the emergentist tradition, are largely decoupled.